| L(s) = 1 | + 1.13·7-s + 20.7·11-s + 38.0·13-s − 4.61·17-s − 46.4·19-s − 17.2·23-s + 138.·29-s + 151.·31-s − 248.·37-s − 92.6·41-s + 443.·43-s + 262.·47-s − 341.·49-s + 278.·53-s − 662.·59-s + 365.·61-s − 786.·67-s + 401.·71-s + 290.·73-s + 23.4·77-s − 185.·79-s − 399.·83-s + 886.·89-s + 43.0·91-s − 48.9·97-s − 1.30e3·101-s − 625.·103-s + ⋯ |
| L(s) = 1 | + 0.0611·7-s + 0.568·11-s + 0.811·13-s − 0.0658·17-s − 0.560·19-s − 0.156·23-s + 0.885·29-s + 0.876·31-s − 1.10·37-s − 0.352·41-s + 1.57·43-s + 0.813·47-s − 0.996·49-s + 0.721·53-s − 1.46·59-s + 0.767·61-s − 1.43·67-s + 0.671·71-s + 0.466·73-s + 0.0347·77-s − 0.263·79-s − 0.528·83-s + 1.05·89-s + 0.0496·91-s − 0.0512·97-s − 1.29·101-s − 0.598·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.341271115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.341271115\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.13T + 343T^{2} \) |
| 11 | \( 1 - 20.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.61T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 17.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 248.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 92.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 443.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 278.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 662.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 365.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 786.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 401.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 290.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 185.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 399.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 886.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 48.9T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.795910886629909621650796288119, −8.300714949177272099272266696584, −7.29106048152123488461403347851, −6.46367621756921392002687762435, −5.84259134996702904548002892701, −4.71566008100769920876416379873, −3.96085387290162478688333560488, −2.97669758053934059022842336785, −1.80567587323228342833205683344, −0.73986334694139655182201659861,
0.73986334694139655182201659861, 1.80567587323228342833205683344, 2.97669758053934059022842336785, 3.96085387290162478688333560488, 4.71566008100769920876416379873, 5.84259134996702904548002892701, 6.46367621756921392002687762435, 7.29106048152123488461403347851, 8.300714949177272099272266696584, 8.795910886629909621650796288119
